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arXiv:cond-mat/0301586 · Physical Review B

One-dimensional conductance through an arbitrary potential

Tobias Stauber

Published Mar 29, 2004• 3 citations
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Abstract

The finite-size Tomonaga-Luttinger Hamiltonian with an arbitrary potential is mapped onto a noninteracting Fermi gas with renormalized potential. This is done by means of flow equations for Hamiltonians and is valid for small electron-electron interaction. This method also yields an alternative bosonization formula for the transformed field operator which makes no use of Klein factors. The two-terminal conductance can then be evaluated using the Landauer formula. We obtain similar results for infinite systems at finite temperature by identifying the flow parameter with the inverse squared temperature in the asymptotic regime.

DOI
10.1103/physrevb.69.113315

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The finite-size Tomonaga-Luttinger Hamiltonian with an arbitrary potential is mapped onto a noninteracting Fermi gas with renormalized potential. This is done by means of flow equations for Hamiltonians and is valid for small electron-electron interaction. This method also yields an alternative bosonization formula for the transformed field operator which makes no use of Klein factors. The two-terminal conductance can then be evaluated using the Landauer formula. We obtain similar results for infinite systems at finite temperature by identifying the flow parameter with the inverse squared temperature in the asymptotic regime.
The conductance of one-dimensional interacting electron systems is calculated in a manner similar to Landauer's argument for non-interacting systems. Unlike in previous studies in which the Kubo formula was used, the conductance is directly evaluated as the ratio of current JJJ to the chemical potential difference ΔμΔμΔμ between right-going and left-going particles. It is shown that both JJJ and ΔμΔμΔμ are renormalized by electron-electron (e-e) interactions, but their ratio, the conductance, is not renormalized at all if the e-e interactions are the only scattering mechanism. It is also shown that nonequilibrium current fluctuation at low frequency is absent in such a case. These conclusions are drawn for both Fermi liquids (in which quasi-particles are accompanied with the backflow) and Tomonaga-Luttinger liquids.
The conductance of one-dimensional interacting electron systems is calculated in a manner similar to Landauer's argument for non-interacting systems.Unlike in previous studies in which the Kubo formula was used, the conductance is directly evaluated as the ratio of current J to the chemical potential difference ∆µ between right-going and left-going particles.It is shown that both J and ∆µ are renormalized by electron-electron (e-e) interactions, but their ratio, the conductance, is not renormalized at all if the e-e interactions are the only scattering mechanism.It is also shown that nonequilibrium current fluctuation at low frequency is absent in such a case.These conclusions are drawn for both Fermi liquids (in which quasi-particles are accompanied with the backflow) and Tomonaga-Luttinger liquids.
Scattering from a localized potential of arbitrary strength is found for weakly interacting electrons. Poor man's renormalization group procedure allows us to calculate the renormalized transmission amplitudes at any energies. Simple formulas that describe the conductance at any temperature are derived. In the presence of the electron-electron backscattering, the low-temperature conductance deviates from the results of Luttinger liquid theory. In particular, the temperature dependence of the conductance may become nonmonotonic. In the presence of a magnetic field, backscattering gives rise to a peak in the differential conductance at bias equal to the Zeeman splitting.
Electron conduction in one-dimensional system is investigated explicitly taking into account the reservoirs to which 1D channel is connected at its ends. We calculate the current driven by the chemical potential difference of two reservoirs, using the linear response theory by Kubo. At 0K, the conductance of 1D channel is expressed in terms of one-electron Green's functions even in the presence of the interaction, in contrast to the theory based on Kubo formula, in which two-electron Green's functions are essentially involved. The formalism in this paper provides a new interpretation of Landauer formula. As regards the interaction effects, the conductance of Fermi liquid without potential scattering is 2 e 2 / h , while that of Tomonaga-Luttinger liquid deceases with power low as the length of 1D channel increases.
We review some of our recent results on the potential scattering in a weakly interacting one-dimensional(1D) electron gas. The technique we developed is a poor man's renormalization group procedure in the scattered wave basis. This technique can treat the renormalizations of the scattering on the barrier and the scattering between the electrons in a coherent way, and it allows us to find the scattering amplitudes on a localized potential of arbitrary strength for electrons at any energy. The obtained phase shifts are used to study the Fermi-edge singularity in an interacting 1D electron system, where anomalous exponent of the power-law singularity in the vicinity of the edge is found. The transmission coefficient is directly related to the conductance of a 1D channel by the Landauer formula. Simple formulas that describe the conductance at any temperature are derived. In spin-[Formula: see text] systems, the electron–electron backscattering induces renormalizations of the interaction constants, which causes the low-temperature conductance to deviate from the results of the Luttinger liquid theory. In particular, the temperature dependence of the conductance may become nonmonotonic. In the presence of a magnetic field, backscattering gives rise to a peak in the differential conductance at bias equal to the Zeeman splitting.